God Between the Numbers, Part 1: Three Kinds of Infinity

If you’re ready to explore, then, let’s play a little numbers game, shall we? No bookies, no mob bosses, and no cops to pay off –just you and me and God makes three.

OK, first question: How many numbers are there between 1 and 2?

If you’re thinking the answer is “none,” think again. You are thinking of integers, also known as counting numbers. Integers are the non-fractional units used to determine the quantity of elements in a finite set. “How many jelly beans in this jar? How many air miles from here to Timbuktu? How many seconds do scientists estimate have elapsed since the Big Bang?” Stuff like that.

Fractional or decimal numbers, on the other hand, are typically used in counting only when parents don’t want to deal with the consequences of prescribed disciplinary measures (“1….2….don’t make me count to 3….2 and a half….2 and three-quarters….”)

But they are vitally important to all other measurements, and they definitely “count” as numbers. For example, 1.3 is a number between 1 and 2, as is 1.34, and 1.3485796468….

So, how many numbers between 1 and 2 again? If you’re starting to think the answer is “unlimited,” you’ve arrived at a fascinating insight about numbers.

The set of all possible integers is what the philosophy of mathematics calls a countable infinity — and for reasons that will become clear, we will also call it an immanent infinity. Picture a horizontal line with 0 at the center, with positive integers spaced evenly apart in progressive fashion on the right, and corresponding negative integers on the left. The line will continue infinitely in both directions, because no matter what integer it reaches, it can always be elongated by adding 1 if it’s a positive number or subtracting 1 if negative. Though the theoretical linear progression never ends, the pattern is established by the integers closest to 0 and it does not deviate, and that makes it a measurable spectrum, similar to linear time.

Decimals are a different story, and this is where it gets interesting. For just as any positive integer can be increased by one, the number of decimal places can be increased. So the set of possible decimal places that can be affixed to any integer is also a countable infinity.

But the set of possible numbers created by any integer’s set of infinite decimal places is itself not countable. There is, for example, no orderly way to progress horizontally from 3.1 to 3.2 if including all possible numeric values between them. You could say that the potential numbers keep increasing vertically on the horizontal spectrum of integers as decimal places are added.

The fact that most such decimal numbers cannot be the result of an algebraic formula makes them “irrational” in mathematical terms, but it doesn’t make them unreal, and they are still potentially useful.

(Among these irrational numbers is another uncountable infinite set called transcendent numbers, which have no discernible end to the quantity of decimal places. The most well-known transcendent number is 𝞹, or “pi,” which is the ratio of a circle’s circumference to its diameter. The golden ratio, observed in many patterns in nature, most commonly in spirals, is also a transcendent number. So, in case you are tempted to find the use of such numbers without practical meaning, imagine a world without perfect circles or nautilus shells.)

Since “almost all” [1] of the numbers in the vertically infinite set of values between two numbers are transcendent, we could call this type of set a transcendent infinity.

Let’s be very clear about something before moving on: there is a transcendent infinity between ANY numbers. Not just integers, but decimal numbers as well. By adding one decimal place to 3.1, for instance, we can make nine possible numbers that are less than 3.2, starting with 3.11. But when we add another decimal place to 3.11, there are nine more numbers smaller than 3.12…and so on.

Posit any two numbers, no matter how relatively close in proximity on any practical spectrum of measurement, and you can use runaway decimal place explosion to create another transcendent infinity, each with its own infinite set of infinities within.

Any. Two. Numbers.

Mind blown yet?

Notice that we have described two different types of infinity, immanent and transcendent, while having yet touched upon what absolute infinity would mean. Obviously an immanent infinity excludes an infinite set of numbers because it only deals in discrete finite values with repeatable patterns. But a transcendent infinity has limitations as well. After all, the infinite numbers between 1 and 2 exclude all numbers below 1 and above 2! To arrive at a conceptualization of absolute Infinity, we must consider an infinite set of all infinite sets, both countable and transcendent, exclusive of no possible numbers.

Absolute Infinity, therefore, is Number itself, or Numerality —the very potential for a numeric value to exist.

Sort of. Read on….

If you are following me to this point, 1) you deserve some kind of medal, and 2) you now have all the tools you need to understand God and your relationship to God as an individual, on three different levels. A Holy Trinity, one might say.

FOOTNOTE [1]: “Almost all” is a fancy mathematical term for the peculiar ratio that results when an infinite set is compared to a finite one. No matter how large the finite set is, it is considered negligible by comparison, while the infinite set never loses its infinitude by being subtracted from or divided.